# What is the mathematical significance of the constant C in an indefinite integral?

As we had seen in an earlier post, calculus bottleneck, calculus presents one of the most difficult topics for the students in higher mathematics. But the problem is not just limited to the students. Teachers feel it too. Too often the emphasis is given on how to solve integration and differentiation problems using “rules” and “methods” while the essence of what is happening is lost. Recently, I asked this question in an interview to a mathematics teacher who was teaching indefinite integration. This teacher had almost a decade of experience in teaching mathematics at +2 level. The teacher tried to answer this question by using an example of the function $$x^{2} + 5$$. Now when we take the derivative of this function, we get

$\dv{ (x^{2} + 5)}{x} = \dv{x^{2}}{x} + \dv{5}{x} = 2x$

as derivative of a constant (5 on our case) is 0). Now the teacher tried to argue, that integration is the reverse of the derivative), so

$\int 2x \, \dd x = \frac{2x^{2}}{2} + C = 2x + C$

After this the teacher tried to argue this $$C$$ represents the constant term (5) in our function $$x^{2} + 5$$. He tried to generalise the result, but he was thinking concretely in terms of the constant in the form of the numbers in the function. The teacher could understand the mechanism of solving the problem, but was not able to explain in clear mathematical terms, why the constant $$C$$ was required in the output of the indefinite integral. This difficulty, I think, partly arose because the teacher only thought in terms solving integrals and derivatives in a particular way, and knew about the connection between the two, but not in a deep way. He did in a way understood the essence of the constant $$C$$, but was not able to understand my question as a general question and hence replied only in terms of concrete functions. Even after repeated probing, the teacher could not get the essence of the question:

why do we add a constant term to the result of the indefinite integral?

To put it in another words, he was not able to generalise an abstract level of understanding from the examples that were discussed. The teacher was thinking only in terms of symbol manipulation rules which are sufficient for problem solving of these types. For example, look at the corresponding rules for differentiation and integration of the function $$x^{n}$$.

$\dv{x^{n}}{x} = n x^{n-1} \iff \int x^{n} \dd x = \frac{x^{n+1}}{n} + C$

Thus, we see according to above correspondence that adding any extra constant $$C$$ to the derivative formula will not affect it. So the teacher claimed it is this constant which appears in the integration rule as well. In a way this is a sort of correct explanation, but it does not get to the mathematical gist of why it is so. What is the core mathematical idea that this constant $$C$$ represents.

Another issue, I think, was the lack of any geometrical interpretation during the discussion. If you ask, what is the geometrical interpretation of the derivative you will get a generic answer along the lines: “It is the tangent to the curve” and for integration the generic answer is along the lines “It is the area under the curve”. Both these answers are correct, but how do these connect to the equivalence above? What is the relationship between the tangent to the curve and area under the curve which allows us to call the integral as the anti-derivative (or is the derivative an anti-integral?). I think to understand these concepts better we have to use the geometrical interpretation of the derivative and the integral from their first definitions.

The basic idea behind the derivative is that it represents the rate of change of a function $$f$$ at a given point. This idea, for an arbitrary function, can be defined (or interpreted) geometrically as:

The derivative of a function $$f$$ at a point $$x_{0}$$ is defined by the slope of the tangent to the graph of the function $$f$$ at the point  $$x = x_{0}$$.

The animation below shows how the slope of the tangent to the sine curve changes. Point $$B$$ in the animation below plots the $$(x, m)$$, where $$m$$ is the slope of the tangent for the given value of $$x$$. Can you mentally trace the locus of point $$B$$? What curve is it tracing?
Now, the tangent to any point on a curve is unique. (Why is it so?) That means if one evaluates a derivative of a function at a point, it will be a unique result for that point.
This being cleared, now let us turn to the indefinite integral. One approach to understanding integration is to consider it as an inverse operation to the derivative, i.e. an anti-derivative.

An anti-derivative is defined as a function $$F(x)$$ whose derivative equals an initial function $$f (x)$$:

$f(x)= \dv{ F(x)}{x}$

Let us take an example of the function $$f(x) = 2x^{2} – 3x$$. The differentiation of this function gives us its derivative $$f'(x) = 4x – 3$$, and its integration gives us anti-derivative.
$F(x) = \frac{2}{3} x^{3} – \frac{3}{2} x^2$

This anti-derivative can be obtained by applying the known rules of differentiation in the reverse order. We can verify that the differentiation of the anti-derivative leads us to the original function.

$F'(x) = \frac{2}{3} 3 x^{2} – \frac{3}{2} 2 x = 2x^{2} – 3x$

Now if add a constant to the function $$F(x)$$, lets say number 4, we get a function $$G(x) = \frac{2}{3} x^{3} – \frac{3}{2} x^2 + 4$$. If we take the derivative of this function $$G(x)$$, we still get our original function back. This is due to the fact that the derivative of a constant is zero. Thus, there can be any arbitrary constant added to the function $$F(x)$$ and it will still be the anti-derivative of the original function $$f(x)$$.

An anti-derivative found for a given function is not unique. If $$F (x)$$ is an anti-derivative (for a function $$f$$ ), then any function $$F(x)+C$$, where $$C$$ is an arbitrary constant, is also an anti-derivative for the initial function because
$\dv{[F(x)+C]}{x} = \dv{ F(x)}{x} + \dv{ C}{x}= \dv{ F(x) }$

But what is the meaning of this constant $$C$$? This means, that each given function $$f (x)$$ corresponds to a family of anti-derivatives, $$F (x) + C$$. The result of adding a constant $$C$$ to any function is that it shifts along the $$Y$$-axis.

Thus what it means for our case of result of the anti-derivative, the resultant would be a family of functions which are separated by $$C$$. For example, let us look at the anti-derivative of $$f (x) = \sin x$$. The curves of anti-derivatives for this function are plotted in will be of the form

$F ( x ) = − \cos x + C$

And this is the reason for adding the arbitrary constant $$C$$ to our result of the anti-derivative: we get a family of curves and the solution is not unique.

Now can we ever know the value of $$C$$? Of course we can, but for this we need to know the some other information about the problem at hand. These can be initial conditions (values) of the variables or the boundary condition. Once we know these we can determine a particular curve (particular solution) from the family of curves for that given problem.

Lev Tarasov – Calculus – Basic Concepts For High Schools (Starts with and explains  the basic mathematical concepts required to understand calculus. The book is in the form of a dialogue between the author and the student, where doubts, misconceptions and aha moments are discussed.)

Morris Kline – Calculus – A physical and intuitive approach (Builds the concepts in the context of the physical problems that calculus was invented to solve.  A book every physics student should read to get an understanding of how mathematics helps solve physical problems.)

Richard  Courant and Fritz John – Introduction to Calculus Analysis (In 2 Volumes) (Standard college level text with in-depth discussions. First volume is rigorous with basic concepts required to conceptually understand the topics and their applications/implications.)

# An example use of H5P item for creating hotspots

Just trying out different H5P items. This item has a flower  and hotspots indicating parts of the flower.

Click on the question marks to know more..

# What is mass renormalization?

A simple explanation of mass renormalization:

The difficulties associated with an infinite rest mass can be successfully overcome in calculation of various effects with the help of mass renormalization which essentially consists in the following. Suppose that it is required to calculate a certain effect and an infinite rest mass appears in the calculations. The quantity obtained as a result of calculations is infinite and is consequently devoid of any physical meaning. In order to obtain a physically reasonable result, another calculation’ is carried out, in which all factors, except those associated with the phenomenon under consideration, are present. This calculation also includes an infinite rest mass and leads to an infinite result. Subtraction of the second infinite result from the first leads to the cancellation of infinite quantities associated with the rest mass. The remaining quantity is finite and characterizes the phenomenon being considered. Thus, we can get rid of the infinite rest mass and obtain physically reasonable results which are confirmed by experiment. Such a method is used, for example, to calculate the energy of an electric field.

– A. N. Matveev, Electricity and Magnetism, p. 16

# Emphasis

emphasis | ˈɛmfəsɪs | noun (plural emphases | ˈɛmfəsiːz | ) [mass noun]

1 special importance, value, or prominence given to something: they placed great emphasis on the individual’s freedom | [count noun] : different emphases and viewpoints

2 stress given to a word or words when speaking to indicate particular importance: inflection and emphasis can change the meaning of what is said

vigour or intensity of expression: he spoke with emphasis and with complete conviction

Emphasis on something means that we want to highlight it from the rest. A common way to do this in text is to italicize or give a boldface or even underline the text. At times colour is added to text to highlight it or colour is added to the background of the text. All these elements of typography work when there is a common background against which these elements standout. Hence emphasise the words as required. But,

## If everything is emphasised, the un-emphasised becomes emphasised.

But consider a block of text which is completely emphasised.

Either the well was very deep, or she fell very slowly, for she had plenty of time as she went down to look about her, and to wonder what was going to happen next.

Either the well was very deep, or she fell very slowly, for she had plenty of time as she went down to look about her, and to wonder what was going to happen next.

Either the well was very deep, or she fell very slowly, for she had plenty of time as she went down to look about her, and to wonder what was going to happen next.

Either the well was very deep, or she fell very slowly, for she had plenty of time as she went down to look about her, and to wonder what was going to happen next.

Either the well was very deep, or she fell very slowly, for she had plenty of time as she went down to look about her, and to wonder what was going to happen next.

Thus we see that the appeal of the emphasis is lost! The only way emphasis will work is to create a background against which it stands out. Let us return to our examples above

Either the well was very deep, or she fell very slowly, for she had plenty of time as she went down to look about her, and to wonder what was going to happen next.

Either the well was very deep, or she fell very slowly, for she had plenty of time as she went down to look about her, and to wonder what was going to happen next.

Either the well was very deep, or she fell very slowly, for she had plenty of time as she went down to look about her, and to wonder what was going to happen next.

Either the well was very deep, or she fell very slowly, for she had plenty of time as she went down to look about her, and to wonder what was going to happen next.

Either the well was very deep, or she fell very slowly, for she had plenty of time as she went down to look about her, and to wonder what was going to happen next.

For me, personally I have not used underline or the highlight. And recently have shifted to coloured italics as my choice of emphasis.

Either the well was very deep, or she fell very slowly, for she had plenty of time as she went down to look about her, and to wonder what was going to happen next.

Sometimes this produces very pretty results (at least I am very happy about them 🙂

(ETBB font with OrangeRed  (#FF4500) italics )

In some cases boldface colour also gives very good results:

Either the well was very deep, or she fell very slowly, for she had plenty of time as she went down to look about her, and to wonder what was going to happen next.

Elements of Typographic Style by Robert Bringhurst

# Escher on Escher

Recently (read some years back) I read Escher on Escher – Exploring the Infinite. This book gives an insight on how Escher viewed himself and his work. How were his social relations with his family, friends and admirers. Fame and appreciation of his work by a wide circle of people came late to Escher, when he was past his 50s. Escher was a perfectionist, he had almost perfected his craft of making woodcuts, taking it to its limits
as far as his hands and eyes could take. But that was the mere mechanical part of his work, the real part was the idea of the graphic print. The ideas it seems struggled a lot in his mind, making the print itself was the easier part. The ideas came to him, but later he strived for something entirely new, and succeeded.

On of the part of the book “Lectures That Were Never Given” has notes from the talks that Escher was supposed to give in the US of Amerika, but could not because of health reasons. Other include translations of his articles that appear in many of the art magazines and journals. These articles tell us how Escher looked at the work he was doing, and his feelings about other artists works. But he was the most critical about his own work.

# 12
But his will and his capacity to produce pictorial images were at the least just as strong as ours. Perhaps even stronger because he was in direct contact with nature, which we usually approach by the way of a cultural and educational system that, if not barring the was, certainly obstructs it for us.
# 13
Illustrations are consequently for the graphic artists (mostly) an indispensable link in the chain of activities, but never his goal. That is probably the reason why a graphic artists cannot suppress a feeling of dissatisfaction when presented with an illustration as end result. You see, I don’t give reasons, only statements.
# 15
The above-mentioned elements of repetition and multiplication is /not/ in conflict with this. On the contrary, order is repetition of units; chaos is multiplicity without rhythm.

Escher also talks about the influence Bach’s music had on his work. He says after hearing Bach’s Goldberg variation in a concert (this is from the acceptance address for an award which he gave to city of Hilversum):

# 20
That was to Bach to whom I have pledged my heart and my intellect at the same time. Such beauty, of composition as well as of execution, cannot be possibly expressed in words.

Maybe same is also true of Escher’s own work. And the inspiration that he got from Bach was at a deep level, maybe if Bach was a contemporary, Escher wouldn’t have felt that lonely in his life. He says:

# 20
Bach’s music may perhaps provide the occasion to say a few words about my work. I had better not expound on the affinity I seem to have discovered between the canon in polyphonic music and the regular division of a plane into figures with identical forms, no matter how striking it is to me that the Baroque composers have performed manipulations with sounds similar to the ones I love to do with visual images.

Allow me to say only that Father Bach has been a strong inspiration to me, and that many a print reached definite form in my mind while I was listening to lucid, logical language he speaks, while I was drinking the clear wine he pours.

And this is what he has to say about his own work:

# 21
I can’t keep fooling around with our irrefutable uncertainties. It is, for example. a pleasure knowingly to mix up two- and three-dimensionalities, flat and spatial, and to make fun of gravity. Are you really sure that a floor can’t also be a ceiling? Are you definitely convinced that you will be on a higher plane when you walk up a staircase? Is it a fact as far as you are concerned that half an egg isn’t also half an empty shell?
Such apparently silly questions I pose first to myself (because I am my own first observer), and then to others who are kind enough to come and observe my work. It is satisfying to note that quite a few people enjoy this kind of playfulness, and that they aren’t afraid to modify their thinking about rock-solid realities.

And about art itself and the feelings that it manifests in us he says:

# 21
To tell you the truth, I find the concept of “art” a bit of dilemma. What one person calls “art” is often not “art” for another. “Beautiful” and “ugly” are old-fashioned concepts that are only rarely brought into the picture nowadays – maybe rightfully so, who is to say? Something repellant, something that gives you a moral hangover, something that hurts your eyes and ear can still be art!

# 21
So I am a graphic artist with heart and soul, but the rating “artist” makes me a feel a little embarrassed.

Next in the book are the “Lectures That Were Never Given”. These are the notes accompanying the slides, the text tells us about the technique and thought and thinking about the work. Escher has interesting way of putting thoughts about his work and the creatures it contains. He suggests to us that the creatures have ideas and behavior of their own.

# 30
On Horsemen and Symmetry Work 67 (glide reflection) The left horsemen, as a creature, was exceptionally obliging and willing. It happens rarely that my subjects so meekly allow themselves to be portrayed in detail.

# 31
While drawing, I feel as if I were a spiritualistic medium, controlled by creatures that I am conjuring up, and it is as if they themselves decide on the shape in which they like to appear.

And the regular division of the plane is a theme which he calls “unusual mania” and its origins.

# 30
I often have wondered at this, for an artist, unusual mania of mine to design periodic drawings. Over the years I made about a hundred fifty of them. In the beginning, that was some forty years ago, I puzzled quite instinctively, apparently without any well-defined purpose. I was simply driven by the irresistible pleasure I felt in repeating the same figures on a piece of paper. I had not yet seen the tile decorations in Al-hambara and never heard of crystallography; so I did not even know that my game was based on rules that have been
scientifically investigated.

And on interpretations of his work, in which people find what they want, religious, spiritualistic and philosophical messages he says this:

# 47 (Reptiles)
I never had any moralizing or symbolizing intention with this print, but some years later one of learned customers told me that it is a striking illustration of the doctrine of reincarnation. So it appears that one can even be symbolizing without knowing it.

And on the self-portraits that he has something to say to us regarding ourselves, ( involving a bit of narcissism, I think) :

# 60 (Hand with Reflecting Sphere, Three Spheres II)
Your own head, or more exactly the point between your eyes, is the center. No matter how you turn or twist yourself, you can’t get out of that central point. You are immovably the focus of your world.

# 61 (Eye)
I choose the features of Good Man Bones, with whom we are all confronted whether we like it or not.

And on Print Gallery, (one of my personal favourites) he says:

# 67 (Print Gallery)
Thus having let our eyes rove in a circular tour around the blank center, we come to the logical conclusion that the young man himself also must be part of print he is looking at. He actually sees himself as a detail of the picture; reality and image are one and the same.

This is something that I can relate to. The life we observe is ultimately the print and we are actually observing ourselves in life as in young man in the Print Gallery. And on perspectives and absurdities of logical opposites he says:

# 73 (Another World)
It may seem absurd to unite nadir, horizon and zenith in one construction, and yet if forms a logical whole.

In form and function the idea of logical opposites forms, much of the basis for Escher’s work. Visit to Al-hambara had a special significance for Escher. Here he found that the Moorish artists had explored the regular division of
the plane, but he lamented that they restricted themselves to abstract geometric forms and not to the anything that is present in nature which he himself felt an urge for.

# 83
After that first Spanish trip in 1922, I became more and more intrigued by the fitting together of congruent figures according to the above-mentioned definition and by the effort to shape this figures in such a way that they would evoke in the observer an association with an object or a living form of nature. (emphasis in original)

# 88
With regard to my present work, this proves to what extent I feel liberated from the graphic arts simply for the sake of graphic arts.

About the graphic artist with which he identified:

# 90
The graphic artist, however, is like a blackbird that sings in the treetop. Again and again he repeats the song, complete in every copy he makes. The more copies people ask him to make, the better he likes it. He hopes the wind will spread his leaves over the Earth, the farther the better – not like the dry leaves in autumn but like feather-light seeds capable of germinating.

And I think Escher has attained this goal which he describes above very well. His works have germinated into new ideas to a new era of graphic artists and others. And on the old techniques of graphic art:

# 90
Consequently, the emphasis falls unjustifiably on process, and one hardly takes into account the actual goal of all that drudgery. No matter how much joy the exercise of a noble craft can bestow, let us not forget that it is a means of repeating and multiplying. Repetition and multiplication – two simple words. The entire world perceivable with the senses will fall apart into meaningless chaos if we could not cling to these two concepts.

On him being called an “expert.”

# 92
A feeling of helplessness comes over me now that I am faced with describing what is meant by this designation. To my unending amazement, however, this is apparently so unusual and in a sense so new that I am unable to identify any “expert” in addition to myself who is sufficiently comfortable with it to give a written explanation.

# 93
By doing this they have opened the gate that gives access to a vast domain, but they themselves have not entered. Their nature is such that they are more interested in the way the gate is opened than in the garden that lies behind it. Sometimes I think I have covered the entire domain and trod all the paths and admired all the views. Then all of a sudden I find another new way, and I taste a new delight.

On his explorations of the plane and its drawings

Because what fascinates me, and what I experience as beauty, is apparently considered dull and dry by others. A plane which one must imagine as extending without boundaries in all directions, can be filled or divided into infinity, according to a limited number of systems, with similar geometric figures that are contiguous on all sides without leaving “empty spaces.”

We don’t to master everything required to construct something in order to appreaciate it.

# 94
Just as I do not consider it necessary to know all the tricks of the graphic trade in order to appreciate prints, neither do I believe that one must master in detail the theoretical fundaments of division of planes in order to learn to value this and to accept that it can exert an inspiring influence, as I have experienced.

On the unending nature of many of creations.

# 95
I see it as a means of representing timelessness, the dimensionlessness, that existed before life commenced and that will return when life again ceases.

On the dynamic nature of his drawings and comparison to film and reading of a book.

# 98
The series of static representations achieved a dynamic character due to the time span that was needed to follow the whole story. In contrast to this cinematographically projected images of a film which appear one in the after the other on an immovable place, onto which viewer’s eye remains directed without moving. In the case of both the medieval story in images and the developing pattern of a regular filling of a plane, the images are located next to each other, and timing becomes a factor in the movements made by viewer’s eye as it follows the story from image to image. In this way, holding a strip of film in hand, one observes it image by image, reading a book is also done more or less in the same manner.

# 99
Is it possible to make a representation of recognizable figures that has no background? To see only a “figure” is not conceivable because something that manifests itself as a figure, that is, as “thing to be seen,” is limited, whether it is real or not. A limitation also means a separation with regard to something else. That “something else” is the background from which the figure (or object sensation) frees itself.

# 100

Imagination and inventiveness, not to mention tenacity, are indispensable for this work. They come to us from “somewhere out there” but we can facilitate their path to us and encourage and
cultivate them in various ways. Among others I found one in writings of Leonardo da Vinchi. This is
the fragment, translated as best I can:

“When you have to represent an image, observe some walls that are besmeared with stains or composed of stones of varying substances. You can discover in them resemblances to a variety of mountainous landscapes, rivers, rocks, trees, vast plains and hills. You can also seen in them battles and human figures, strange facial features and items of clothing, and an infinite number of other things whose forms you can straighten out and improve. It is the same with crumbling walls as it is with the sound of church bells, in which you can discover every name and every word you want.”

Despite all this conscious and personal effort the illustrator still gets the feeling that some kind of magic action is taking place as he moves his lead pencil over the paper. It seems as if it isn’t he who determines the shapes but rather that the simple, flat stain is guiding or impeding the movements of the hand that draws, as if the illustrator were a spiritualistic medium. In fact, he is amazed, not to say taken aback, at what he sees appearing under his hand, and he experiences with regard to his creations a humble feeling of gratitude or of resignation depending on whether they behave willingly or reluctantly.

# EM Spectrum in Astronomy

EM Spectrum in Astronomy from Astrobites

I created a mindmap from the information above

λ > 1 mm
ν < 300 GHz

• Objects
• AGN JEts
• Supernovae
• Tidal Disruption Events
• H II regions
• Gamma ray bursts
• Processes
• Observation
• Ground based
• Telescopes
• Green Bank Telescope (GBT)
• Five-hundred-meter Aperture Spherical Telescope (FAST)
• Very Large Array (VLA)
• Square Kilometer Array (SKA)
• Low-Frequency Array (LoFAR),
• Giant Meterwave Radio Telescope (GMRT)
• Microwave/Sub-mm
λ ~ 300 μm to 1 mm
ν ~ 1 THz to 300 GHz

• Objects
• CMB
• High energy phenomena
• Relativistic jets
• Cold dust
• Cold gas
• Galaxies at high z
• Processes
• Observation
• Space
• Ground
• Telescopes
• Space
• Cosmic Background Explorer (COBE)
• Wilkinson Microwave Anisotropy Probe (WMAP)
• Planck
• Ground
• Submillimeter Array (SMA
• Atacama Large Millimeter/submillimeter Array (ALMA).
• Infrared
λ ~ 300 μm to 2.5 μm
ν ~ 1 THz to 120 THz

• Far-Infrared
λ ~ 15 μm to 300 μm
ν ~ 20 THz 1 THz

• Objects
• Cool dust
• Cool gas
• star forming galaxies
• young stellar objects
• proto-stars
• pre-main sequence stars
• Processes
• Observation
• Space
• Telescopes
• Infrared Astronomical Satellite (IRAS)
• Infrared Space Observatory (ISO)
• Herschel
• Mid-Infrared
λ ~ 2.5 μm to 15 μm
ν ~ 120 THz 20 THz

• Objects
• Cosmic dust
• surrounding young stars
• protoplanetary disks
• zodiacal dust
• Solar system objects
• planets
• comets
• asteroids
• Processes
• Observation
• Space
• Ground
• Telescopes
• Ground
• Infrared Telescope Facility (IRTF)
• United Kingdom Infrared Telescope (UKIRT)
• Space
• James Webb Space Telescope
• Wide-field Infrared Survey Explorer (WISE)
• Spitzer
• Near-Infrared
λ ~ 0.8 μm to 2.5 μm
ν ~ 380 THz 120 THz

• Objects
• M-dwarfs
• Cool stars
• Low-mass stars
• Galaxies
• Processes
• Observation
• Space
• Ground
• Telescopes
• Ground
• 2MASS survey
• Infrared Telescope Facility (IRTF)
• United Kingdom Infrared Telescope (UKIRT)
• Visible and Infrared Survey Telescope for Astronomy (VISTA)
• Space
• James Webb Space Telescope
• Optical
λ ~ 350 nm to 800 nm
ν ~ 860 THz 380 THz

• Objects
• Ionized gases
• Stars
• Galaxies
• Processes
• Non-thermal
• Observation
• Both Ground and Space
• Telescopes
• Ground
• W.M. Keck telescopes
• Very Large Telescopes
• Southern African Large Telescope (SALT)
• Space
• Hubble Space Telescope
• Gaia
• Kepler
• Transiting Exoplanet Survey Satellite (TESS).
• Ultra-violet
λ ~ 10 nm to 350 nm
ν ~ 3e16 Hz to 860 Hz
E ~ 120 eV to 3.5 eV

• Objects
• Thermal
• O Stars
• B Stars
• white dwarfs
• Non-thermal
• AGN (continuous emission)
• Processes
• Non-thermal sources
• Observation
• Ground (longest)
• Space
• Telescopes
• AstroSAT (2015)
• Galaxy Evolution Explorer (GALEX) (2003)
• Hubble Space Telescope (1990)
• Neil Gehrels Swift Observatory (2004)
• X-Ray
λ ~ 10 pm to 10 nm
ν ~ 3e19 Hz to 3e16 Hz
E ~ 120 keV to 0.12 keV

• Objects
• X-Ray binaries
• AGN
• Neutron stars
• Processes
• Thermal Emission
• Free-Free emission
• Accretion
• Observation
• From Space
• Telescopes
• Uhuru (1973)
• Einstein (1978-81)
• ROSAT (1999)
• Chandra (1999)
• XMM-Newton (1999)
• NuSTAR (2012)
• eROSITA (2019)
• Gamma-Ray
λ < 10 pm
ν > 3e19 Hz
E > 120 keV

• Objects
• AGN with Relativistic Jets
• Gamma Ray Binaries
• Gamma Ray Bursts
• Processes
• Gamma Decay
• Pair-Annihilation
• Shock Waves
• Inverse-Compton Scattering
• Observation
• From Space
• Telescopes
• Compton Gamma-ray Observatory (1991)
• International Gamma-Ray Astrophysics Laboratory (INTEGRAL) (2002)
• Fermi (2008)

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# Mysterious spiral shells

I have visited one of the most iconic sea forts in Maharashtra – the Sindhudurga at Malvan on the Konkan Coast twice. It is one of the most beautiful sea forts you will witness. The crystal clear waters and blue skies will be imprinted in your memories. The first time I had an analog camera with me, so I couldn’t take as many photos as I would have wanted (one of the analogue snaps is above c. 2001). The second time I had a really nice digital camera with me, so I did take a load of photos (all the ones below).

Maratha navy was a very formidable force on the Konkan coast. The maratha navy under the Angre’s was a force feared by the Portuguese and English alike. They ruled the waters from North of Goa to Colaba Fort (not the same as Colaba in South Bombay) with their capital at Gheria (Vijaydurg), in the later half of 1700s. Maratha navy ships were fastest and very agile in open waters and were absolute terror for the europeans. Anticipating the need for a strong navy as well, Sindhudurga was one of the first sea forts to be built by Shivaji. The fort is built on a rocky outcrop off the coast of Malvan. The fort played an important role after Shivaji as well. It housed Tarabai, the widower queen of younger son Raja Ram during the invasion Mughals. The fort has walls which have stood the test of time and are still standing well even after several centuries against sea water and constant barrage of waves. (though in the second visit we found parts of wall had collapsed). The architecture of the fort is amazing, though we only see the bastions and ramparts now. The fort walls are shaped smoothly when required (in the mathematical sense of continuity), unlike later european forts which are more angular in nature. The front gate is hidden inside a curved pathway between two bastions so that is not visible directly (hence cannot be hit directly with a canon. This mechanism is also seen in several other forts such as Raigad and Janjira. But so much about the fort, coming back to the topic of the post.

On the southern end of the fort there is a small door which opens to a patch of beach. This is one of the best spots on the fort if you are a nature lover. In this small patch there is clear water and beach.

The beach sand is coarse, meaning most of it is actually made of seashells. This is a treat to see, myriad shapes, sizes and colours of seashells blended to form the sand. Sand is a fascinating mixture which results from erosion (air, water) and evolution over long time scales. Sand has both inorganic and organic content.

The sea shells are logarithmic in nature, with the nautilus perhaps being seen as exemplary. In many cases of shells the logarithmic spiral is obvious (see the photo), but in other cases as well it shows logarithmic growth.

See a previous post on patterns in nature, this post is a sort of extension of that. For an extensive and excellent treatise on spirals see The Curves of Life by Theodore Cook. On a side note, you should also read Junijo Ito’s spiral themed horror manga Uzumaki.

I did collect a few noteworthy shells from this patch (there were so many to collect!), and these included a few approximately hemispherical ones which were flat on the other side. These shells had a nice logarithmic spiral on the flat side which was also relatively smooth. While the hemispherical side was comparatively rougher.

Now in all other shells, one could easily visualise where would the animal be that created the shell or how the animals used that shell. for example

But in case of these spirals I could not understand how or where the animal would be using this shell. There was no hole to hide or provide a protection to the animal, as it was solid. It was indeed a puzzle. The shells definitely belonged to an animal, but which one and how did it use it? I asked around but did not get any clear answers. Thus began a journey to unravel the mystery. The image search for spiral shell did not help, though I came to know that these shells are used in astrology oriented rings with silver. And apparently they are seen in jewellery shops.

The first thing we knew for sure was that the shell belonged to a marine animal. But how did that animal use this? It remained an unsolved question for a couple of years. I would ask around to anyone with some knowledge of zoology, but it didn’t get any answers. Then one day someone did faintly recognised, “Isn’t this an operculum?” Now a quick image search with this new term, operculum, I learned gave the answer to this mystery. And commonly it is also known as cat’s eye shell. And the long mystery was solved:

The operculum is attached to the upper surface of the foot and in its most complete state, it serves as a sort of “trapdoor” to close the aperture of the shell when the soft parts of the animal are retracted. The shape of the operculum varies greatly from one family of gastropods to another. It is fairly often circular, or more or less oval in shape. In species where the operculum fits snugly, its outline corresponds exactly to the shape of the aperture of the shell and it serves to seal the entrance of the shell. (wiki)

TODO: Identify all the shells in the photos..

## References

Deepak Apte – The Book of Indian Shells (BNHS)

# Free Graphics Illustration Resources and Repositories

Finding a good and accurate graphic or illustration for your need is something that we all struggle with. On top of that if the requirement is that the graphic resource has to be free (as in freedom) then it further narrows down the options. Sometimes you see the graphic you want, but its license terms are unknown or are not agreeable to your work as they are not released freely. So what do you do? Either you use an illustration which is not perfect fit or you use one which breaks your resource (in terms of license).  But the problems is also that there are many free resource repositories which are very well not known. I have personally come across great many graphic resources, only to be forgotten after their current need is finished. This post is an attempt to overcome this. This post is a collection of various free graphics and illustration resources and repositories that I have found useful over the years. It is also a sort of personal bookmark list of the these resources if and when I need them in the future. I hope this will be of use to others too. I will keep on updating this list with new resources that I find. If you know of any resources which are missing please do post in the comments.

The Internet Archive Image Search https://archive.org/details/image

Wikimedia Commons https://commons.wikimedia.org/

Open source illustrations kit https://illlustrations.co/

NYPL Public Domain Archive https://nypl.getarchive.net

David Rumsey Map Collection  https://www.davidrumsey.com/  also https://archive.org/details/david-rumsey-map-collection

https://archive.org/details/bibliothequesaintegenevieve_image

Unsplash Free Images

NASA Images

ESO Images

A library of microorganisms https://archive.org/details/cmpuj

National Gallery of Art also https://archive.org/details/national-gallery-of-art-images

Flickr Collections (there are several collections on Flickr which are openly licensed)

Flickr Commons

NOAA Photo Library https://flickr.com/photos/noaaphotolib/

# On Stories

What is a story?

A story is a series of imagined sequence of events which may or may not be based on true events. The story takes place somewhen, somewhere. A story may take place in the past, or the present or the future. A story may take place in a real place, or an imaginary one (even the surface of a neutron star!). A story has characters and a narrator (can there be a story without either?). Sometimes the characters tell us the story. Sometimes the narrator tells us the story. Some stories are fun, some are dark, others are just boring. A good story is like a fishing hook. It will compel the listener, and keep them eager for what happens next. It is the anticipation of events, which makes a story interesting. What good would be a story if you knew exactly what is going to happen? They say don’t let facts ruin a good story. But facts themselves can make a great story. Reality is stranger than fiction. Reality is a fractal. More you look deeper you see. The mundane becomes the mysterious. Everyday thing becomes enigmatic.

Timeless stories

There are stories which endure time. They were perhaps told since humans acquired language. Perhaps the enormous capacity of language to express imagination was developed to tell the stories. Stories are culture. Stories give us identity. Stories give us rules and norms. Stories make us who we are. Stories tell us about gods, about demons, about magical beings, about kings, about heaven and hell.

Visual Stories

Not all stories have words. Some stories are told via images. Before the written word, images were the only permanent thing. The spoken word is ephemeral, while images endure. Spoken word is for people who are present then and there, the image can transcend time and distance. The images can be somewhen, somewhere other than where and when it was created. Spoken words live only in the present.

Variations on a Theme

Stories change. When a story is told from one person to another, while retelling the person will inadvertently make changes. Though core idea of the story may remain same, some of the details may change. Thus stories multiply. Language is not a barrier. Same story can be told in different languages. Characters, places may change but the stories remain the same. It is like a person wearing different outfits suitable for the local environment.

Storytellers

But how do stories get made? Who makes them? Well, we make stories.

Each one of us has stories to tell. Some people are good at telling them, some aren’t. The good stories stick around. They pass on. From person to person, from generation to generation. From parents to children. From elders to youth. From friends to friends. From teachers to students. Storytellers form a network. Stories link them to those who came before them and those who come after them.

Emotions

Stories evoke emotions. Some make us laugh, some make us cry. Some us happy, some make us sad. Some take us on adventures we will never be able to go. Some make us confront our deepest fears and invoke terror in us. Some make us feel elated, and inspire us to do things. Some possess wisdom, some are just folly. Some stories are about mundane things. Some stories are about mysteries. Some stories you can readily identify with, some are so alien to our experience.

Stories are Everything

Without stories we are nothing. We relate to others via stories in which we are the characters. Everyone has at least one story to tell, their own story. Will your life be a story worth telling others?

# Change and hiccups

The site was down for a while due to a technical hiccup while transfering from WP to AWS. But its up and running now, I guess change and transformation is a part of the process… So will start with regular posts, lets see at least once a week!

PS: thanks to @imsmubarak for helping setup the AWS and @alpesh for sorting the DNS and domain transfer issue…